#set document(title: "7.5 Standard Normal", author: "OpenStax") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 7.5#h(0.6em)Standard Normal #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Prerequisites] Effects of Linear Transformations, Introduction to Normal Distributions #linebreak() #linebreak() ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Learning Objectives] + State the mean and standard deviation of the standard normal distribution + Use a Z table + Use the normal calculator + Transform raw data to Z scores ] As discussed in the introductory section, normal distributions do not necessarily have the same means and standard deviations. A normal distribution with a mean of 0 and a standard deviation of 1 is called a #strong[standard normal distribution]. Areas of the normal distribution are often represented by tables of the standard normal distribution. A portion of a table of the standard normal distribution is shown in Table 1. Table 1. A portion of a table of the standard normal distribution. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Z], [Area below]), [-2.5], [0.0062], [-2.49], [0.0064], [-2.48], [0.0066], [-2.47], [0.0068], [-2.46], [0.0069], [-2.45], [0.0071], [-2.44], [0.0073], [-2.43], [0.0075], [-2.42], [0.0078], [-2.41], [0.008], [-2.4], [0.0082], [-2.39], [0.0084], [-2.38], [0.0087], [-2.37], [0.0089], [-2.36], [0.0091], [-2.35], [0.0094], [-2.34], [0.0096], [-2.33], [0.0099], [-2.32], [0.0102], )) The first column titled "Z" contains values of the standard normal distribution; the second column contains the area below Z. Since the distribution has a mean of 0 and a standard deviation of 1, the Z column is equal to the number of standard deviations below (or above) the mean. For example, a Z of -2.5 represents a value 2.5 standard deviations below the mean. The area below Z is 0.0062. The same information can be obtained using the following Java applet. Figure 1 shows how it can be used to compute the area below a value of -2.5 on the standard normal distribution. Note that the mean is set to 0 and the standard deviation is set to 1. #figure(figph[Screenshot of the normal-distribution applet set to the STANDARD normal — Mean: 0, Sd: 1 — on an axis from -4 to 4, with the Below option selected and -2.5 entered. Only the small sliver of the left tail below -2.5 is shaded, and the result reads Shaded area: 0.006210.], alt: "Screenshot of the normal-distribution applet set to the STANDARD normal — Mean: 0, Sd: 1 — on an axis from -4 to 4, with the Below option selected and -2.5 entered. Only the small sliver of the left tail below -2.5 is shaded, and the result reads Shaded area: 0.006210.", caption: [Figure 1. An example from the applet.]) Calculate Areas A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the following formula: Z = (X - μ)/σ where Z is the value on the standard normal distribution, X is the value on the original distribution, μ is the mean of the original distribution, and σ is the standard deviation of the original distribution. As a simple application, what portion of a normal distribution with a mean of 50 and a standard deviation of 10 is below 26? Applying the formula, we obtain Z = (26 - 50)/10 = -2.4. From Table 1, we can see that 0.0082 of the distribution is below -2.4. There is no need to transform to Z if you use the applet as shown in Figure 2. #figure(figph[Screenshot of the same applet set to Mean: 50, Sd: 10 with Below 26 entered. The shaded left-tail sliver and the answer, Shaded area: 0.008198, are what you get without converting to a Z score first — 26 is 2.4 standard deviations below 50, and 0.0082 of the distribution lies below Z = -2.4.], alt: "Screenshot of the same applet set to Mean: 50, Sd: 10 with Below 26 entered. The shaded left-tail sliver and the answer, Shaded area: 0.008198, are what you get without converting to a Z score first — 26 is 2.4 standard deviations below 50, and 0.0082 of the distribution lies below Z = -2.4.", caption: [Figure 2. Area below 26 in a normal distribution with a mean of 50 and a standard deviation of 10.]) If all the values in a distribution are transformed to Z scores, then the distribution will have a mean of 0 and a standard deviation of 1. This process of transforming a distribution to one with a mean of 0 and a standard deviation of 1 is called #strong[standardizing] the distribution.